Attracting current and equilibrium measure for attractors on ℙ<Superscript><Emphasis Type="Italic">k</Emphasis></Superscript>

Let f be a holomorphic endomorphism of ℙ ^k having an attracting setA. We construct an attracting current and an equilibrium measure associated toA. The attracting current is weakly laminar and extremal in the cone of invariant currents. The equilibrium measure is mixing and has maximal entropy onA.     

Suites d’Applications Méromorphes Multivaluées et Courants Laminaires

Let F_n: X₁ → X₂ be a sequence of (multivalued) meromorphic maps between compact Kähler manifolds. We study the asymptotic distribution of preimages of points by F_n and, for multivalued self-maps of a compact Riemann surface, the asymptotic distribution of repelling fixed points. Let (Z_n) be a sequence of holomorphic images of ?^s in a projective manifold. We prove that the currents, defined by integration on Z_n, properly normalized, converge to currents which satisfy some laminarity property. We also show this laminarity property for the Green currents, of suitable bidimensions, associated to a regular polynomial automorphism of ?^k or an automorphism of a projective manifold.     

Courants extrémaux et dynamique complexe

We construct extremal positive closed currents of any bidegree on the complex projective space Pk, which are not current of integration along irreducible analytic subsets. We apply these results to the dynamical study of some polynomial endomorphisms of Ck, for which we construct an ergodic measure of maximal entropy.     

Invariant currents and dynamical Lelong numbers

Let ƒ be a polynomial automorphism of ℂ^k of degree λ, whose rational extension to ℙ^k maps the hyperplane at infinity to a single point. Given any positive closed current S on ℙ^k of bidegree (1,1), we show that the sequence λ⁻ⁿ(ƒⁿ)*S converges in the sense of currents on ℙ^k to a linear combination of the Green current T₊ of ƒ and the current of integration along the hyperplane at infinity. We give an interpretation of the coefficients in terms of generalized Lelong numbers with respect to an invariant dynamical current for ƒ⁻¹.     

Legendrian tori and the semi-classical limit

Let X be CPn or a compact smooth quotient of the n-dimensional complex hyperbolic space, n>1. Let L be a hermitian holomorphic line bundle (with hermitian connection) on X chosen as follows: if X=CPn then L is the hyperplane bundle, and in the second case L is chosen so that L⊗(n+1)=KX⊗E, where KX is the canonical line bundle and E is a flat line bundle. The unit circle bundle P in L^∗ is a contact manifold. Let k^′ be a fixed positive integer. We construct certain Legendrian tori in P (the construction depends, in particular, on the choice of k^′) and sequences {uk}, k=k^′m, , of holomorphic sections of L⊗k associated to these tori. We study asymptotics of the norms ‖ukk_‖ as m→+∞ and, in particular, apply this result to construct explicitly certain non-trivial holomorphic automorphic forms on the n-dimensional complex hyperbolic space. We obtain an n>1 analogue of the classical period formula (this is a well-known statement for automorphic forms on the upper half plane, n=1).     

The completeness of a polynomial vector field is determined by a transcendental trajectory

We prove that every polynomial vector field on C² that is complete on a transcendental (proper and non-algebraic) trajectory is complete in C².     

Determination of formal CR mappings by a finite jet

We prove that if M is a connected real-analytic holomorphically nondegenerate hypersurface in Cn+1, then for any point p∈M there exists an integer k such that any two germs at p of local biholomorphic mappings that send M into itself and whose k-jets agree at p are identical.The above is a special case of a more general theorem stated for formal hypersurfaces that gives a finite jet determination result for the class of formal mappings whose Jacobian determinant does not vanish identically.     

Polynomial diffeomorphisms of C2: V. critical points and Lyapunov exponents

There is an invariant measure μ, which is the pluri-complex version of the harmonic measure of the Julia set for polynomial maps of C.In this paper we give an integral formula for the Lyapunov exponents of a polynomial automorphism with respect to μ, analogous to the Brolin-Manning formula polynomial maps of C.Our formula relates the Lyapunov exponents to the value of a Green function at a type of critical point which we define in this paper. We show that these the critical points have a natural dynamical interpretation.     

Another class of balanced graph designs: balanced circuit designs

A block considered as a set of elements together with its adjacency matrix A is called a C-block if A is the adjacency matrix of a circuit. A balanced circuit design with parameters v, b, r, k, λ (briefly, BCD(v, k, λ)) is an arrangement of v elements into bC-blocks such that each C-block contains k elements, each element occurs in exactly rC-blocks and any two distinct elements are linked in exactly λ C-blocks.We investigate conditions for the existence of BCD and show, in particular, that if the block-size k ? 6 and the trivial necessary conditions are satisfied, then the corresponding BCD exists.     

Drift transforms and Green function estimates for discontinuous processes

In this paper, we consider Girsanov transforms of pure jump type for discontinuous Markov processes. We show that, under some quite natural conditions, the Green functions of the Girsanov transformed process are comparable to those of the original process. As an application of the general results, the drift transform of symmetric stable processes is studied in detail. In particular, we show that the relativistic α-stable process in a bounded C^1,1-smooth open set D can be obtained from symmetric α-stable process in D through a combination of a pure jump Girsanov transform and a Feynman-Kac transform. From this, we deduce that the Green functions for these two processes in D are comparable.     

Dynamics of polynomial automorphisms ofC k

We study the dynamics of polynomial automorphisms ofC ^k . To an algebraically stable automorphism we associate positive closed currents which are invariant underf, consideringf as a rational map onP ^k . These currents give information on the dynamics and allow us to construct a canonical invariant measure which is shown to be mixing.     

Generic linear systems for projective CR manifolds

For compact CR manifolds of hypersurface type which embed in complex projective space, we show that for all k large enough there exist linear systems of O(k) which when restricted to the CR manifold are generic in a suitable sense. These systems are constructed using approximately holomorphic geometry.     

L'équation de Poisson et les noyaux de Green associés à un opérateur différentiel singulier sur R+

We consider a class ℳ of singular differential operators on the half line and ⋆ the convolution on ℝ₊. associated with L ε ℳ. If μ(≠ ɛ₀) is a probability measure on ℝ₊, we study the asymptotic behaviour of the solution of both Poisson equations Lu = −ƒ and (μ. − ɛ₀) ⋆ u = −ƒ where ƒ ε C_k(ℝ₊) is given. The results follow from a more general study on the precise asymptotic behaviour of the Green kernel of the convolution semigroups associated with L.     

Super-rigidity for CR embeddings of real hypersurfaces into hyperquadrics

We consider holomorphic mappings sending a given Levi-nondegenerate pseudoconcave hypersurface M in Cn+1 into a nondegenerate hyperquadric of the same signature in PCN+1 and show that if M is sufficiently close to a hyperquadric in a certain sense, then any two such mappings differ only by an automorphism of the hyperquadric.     

Regularity of finite-dimensional realizations for evolution equations

We show that a continuous local semiflow of C^k-maps on a finite-dimensional C^k-manifold M with boundary is in fact a local C^k-semiflow on M and can be embedded into a local C^k-flow around interior points of M under some weak assumption. This result is applied to an open regularity problem for finite-dimensional realizations of stochastic interest rate models.     

Cohomology rings and formality properties of nilpotent groups

We analyze k-stage formality and relate resonance with this type of formality properties. For instance, we show that, for a finitely generated nilpotent group that is k-stage formal, the resonance varieties are trivial up to degree k. We also show that the cohomology ring, truncated up to degree k+1, of a finitely generated nilpotent, k-stage formal group is generated in degree 1; this criterion is necessary and sufficient for a finitely generated, 2-step nilpotent group to be k-stage formal. We compute resonance varieties for Heisenberg-type groups and deduce the degree of partial formality for this class of groups.     

On stability and continuity of bounded solutions of degenerate complex Monge-Ampère equations over compact Kähler manifolds

We obtain a stability estimate for the degenerate complex Monge-Ampère operator which generalizes a result of Ko?odziej (2003) [12]. In particular, we obtain the optimal stability exponent and also treat the case when the right-hand side is a general Borel measure satisfying certain regularity conditions. Moreover, our result holds for functions plurisubharmonic with respect to a big form, thus generalizing the Kähler form setting in Ko?odziej (2003) [12]. Independently, we also provide more detail for the proof in Zhang (2006) [18] on continuity of the solution with respect to a special big form when the right-hand side is Lp-measure with p>1.     

On diffusivity of a tagged particle in asymmetric zero-range dynamics

Consider a distinguished, or tagged particle in zero-range dynamics on Zd with rate g whose finite-range jump probabilities p possess a drift ∑jp(j)≠0. We show, in equilibrium, that the variance of the tagged particle position at time t is at least order t in all d?1, and at most order t in d=1 and d?3 for a wide class of rates g. Also, in d=1, when the jump distribution p is totally asymmetric and nearest-neighbor, and the rate g(k) increases, and g(k)/k either decreases or increases with k, we show the diffusively scaled centered tagged particle position converges to a Brownian motion with a homogenized diffusion coefficient in the sense of finite-dimensional distributions. Some characterizations of the tagged particle variance are also given.     

Involutions of Type G<Subscript>2</Subscript> Over Fields of Characteristic Two

We continue a study of automorphisms of order 2 of algebraic groups. In particular we look at groups of type G₂ over fields k of characteristic two. Let C be an octonion algebra over k; then Aut(C) is a group of type G₂ over k. We characterize automorphisms of order 2 and their corresponding fixed point groups for Aut(C) by establishing a connection between the structure of certain four dimensional subalgebras of C and the elements in Aut(C) that induce inner automorphisms of order 2. These automorphisms relate to certain quadratic forms which, in turn, determine the Galois cohomology of the fixed point groups of the involutions. The characteristic two case is unique because of the existence of four dimensional totally singular subalgebras. Over finite fields we show how our results coincide with known results, and we establish a classification of automorphisms of order 2 over infinite fields of characteristic two.     

Dynamical systems which realize given random bi-sequences of points on their orbits

A dynamical system consists of a phase space of possible states, together with an evolution rule that determines all future states and all past states given a state at any particular moment. In this paper, we show that for any countable random infinite bi-sequences of states of some phase space, there exists an evolution rule in C⁰-topology which realizes precisely the given sequences of states on their orbits and satisfies some regular conditions on the times to realize the states.